Pixelated K-edge coded aperture system for compressive spectral X-ray imaging

ABSTRACT

Aspects of the invention are directed to systems and methods for generating spectral computed tomography data for spectral X-ray image reconstruction using of pixelated k-edge apertures. A method is provided for generating a spectral computed tomography. The method includes the steps of generating a plurality of X-ray beams; encoding the plurality of X-ray beams by transmitting the plurality of beams through a pixelated K-edge coded aperture structure; detecting the encoded plurality of X-ray beams; and reconstructing a spectral CT image from the encoded plurality of X-ray beams.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims priority benefit of a national stage applicationunder 35 U.S.C. 371 of International Application No. PCT/US2018/040310,filed Jun. 29, 2018, which is related to and claims priority to U.S.Provisional Application No. 62/526,389 entitled “Pixelated K-Edge CodedAperture System for Compressive Spectral X-Ray Imaging” filed on Jun.29, 2017, the disclosures of each of these applications beingincorporated herein by reference in their entireties for all purposes.

FIELD OF THE INVENTION

Aspects of the invention are directed to systems and methods forgenerating spectral computed tomography data for X-ray spectral imagereconstruction using a pixelated K-edge coded aperture structure.

BACKGROUND

X-ray computed tomography (hereafter “CT”) has become an important toolin medical diagnosis. Tomographic gray scale images obtained fromconventional systems, however, are often insufficient to revealdifferences between materials having different chemical compositions butthe same X-ray attenuation coefficients. Spectral Computed Tomography(hereafter “SCT”) not only provides morphological information, asconventional CT scans do, but it also allows material decomposition aswell. Thus, the emerging field of SCT has important application inmedical imaging and transit security.

On the other hand, higher energy resolution can be achieved by usingphoton counting detectors, which can identify the energy of incomingphotons and record the data in the corresponding energy bins. Theadvantages of spectral tomography, however, are hindered by severaltechnical challenges that prevent their broad practical application,such as costly photon counting detectors and their low signal to noiseratio. In addition, traditional methods for spectral X-ray CT imagingare time intensive as they use multiple scans when employing standardintegrating detectors.

Accordingly, there is a need for improved methods and system forgenerating spectral computed tomography data.

SUMMARY OF THE INVENTION

Aspects of the invention are directed to systems and methods forgenerating spectral computed tomography data for X-ray spectral imagereconstruction using a pixelated K-edge coded aperture structure.

According to one aspect of the invention, a method is provided forgenerating spectral computed tomography data. The method includes thesteps of generating a plurality of X-ray beams; encoding the pluralityof X-ray beams by transmitting the plurality of beams through apixelated K-edge coded aperture structure; detecting the encodedplurality of X-ray beams; and reconstructing a spectral CT image fromthe encoded plurality of X-ray beams.

In accordance with a further aspect of the invention, another method isprovided for generating spectral X-ray tomography data. The methodincludes the steps of scanning a target with a plurality of X-ray beamsduring at least one pass of an X-ray beam generator with respect to atarget and encoding the plurality of X-ray beams during the at least onepass by transmitting the plurality of X-ray beams through a pixelatedK-edge coded aperture structure. The pixelated K-edge coded aperturestructure delineates a plurality of openings. The method furtherincludes the steps of detecting an intensity of the encoded plurality ofX-ray beams from the at least one pass and reconstructing a spectral CTimage of the target from the encoded plurality of X-ray beams.

According to another aspect of the invention, a system employing apixelated K-edge coded aperture structure is provided for generatingspectral computed tomography data. The system includes at least oneX-ray generator configured for producing a plurality of X-ray beams andat least one pixelated K-edge coded aperture structures delineating aplurality of openings. The plurality of openings are associated with atleast one K-edge filter, such that the difference of the spectra of theplurality of X-ray beams transmitted through the plurality of openingshas an energy band corresponding to a difference between K-edge valuesof a corresponding balanced pair of K-edge filters. The system alsoincludes at least one detector configured to detect the plurality ofX-ray beams transmitted through the at least one pixelated K-edge codedaperture structure.

BRIEF DESCRIPTION OF THE DRAWINGS

The invention is best understood from the following detailed descriptionwhen read in connection with the accompanying drawings, with likeelements having the same reference numerals. It is emphasized thataccording to common practice the various features of the drawings arenot drawn to scale unless otherwise indicated. On the contrary, thedimensions of the various features may be expanded or reduced forclarity. Included in the drawings are the following figures.

FIG. 1 is a schematic depicting a system employing pixelated k-edgeapertures for generating a spectral computed tomography in accordancewith aspects of the invention;

FIG. 2A is a schematic depicting a pixelated K-edge coded structureformed from a single structure having a plurality of openings accordingto aspects of the invention;

FIG. 2B is a schematic depicting a system using the pixelated K-edgecoded aperture mask of FIG. 2A;

FIG. 2C is a graph of the filtered spectra produced using the system ofFIG. 2B with a particular set of Ross filters;

FIG. 2D is a schematic depicting a pixelated K-edge coded structureformed from a first structure for pixelating the X-ray beams and asecond structure for filtering the X-ray beams that is separate from thefirst structure according to aspects of the invention;

FIG. 3 is a method of generating spectral computed tomography data inaccordance with aspects of the invention;

FIG. 4 is a schematic depicting exemplary steps for obtaining spectralCT images according to aspects of the invention;

FIG. 5 is a schematic depicting further non-limiting, exemplary stepsfor obtaining spectral CT images in accordance with aspects of theinvention;

FIG. 6 is a schematic depicting additional non-limiting, exemplary stepsfor obtaining spectral CT images according to aspects of the invention;

FIG. 7 depicts a spectral CT imaging system having an X-ray generatorthat produces fan beams and a X-ray line detector in accordance withaspects of the invention;

FIG. 8 depicts a non-limiting embodiment of a spectral CT imaging systemhaving an X-ray generator that produces cone beams and a X-ray twodimensional detector according to aspects of the invention;

FIGS. 9A and 9B depict non-limiting embodiments of spectral CT imagingsystems employing a conveyor belt in accordance with aspects of theinvention;

FIGS. 10A-10E depict non-limiting embodiments of spectral CT imagingsystems according to aspects of the invention;

FIG. 11 is a schematic depicting a process to obtain aquasi-monochromatic X-ray spectrum using a Ross pair formed fromdysprosium and cerium according to aspects of the invention.

FIG. 12 is a schematic of a method for using a compressive spectralX-ray imaging (CSXI) system for a fan beam architecture, where two X-raybeam shots are projected through different materials in accordance withaspects of the invention;

FIG. 13 is a schematic illustrating the coded aperture matrices for asystem according to aspects of the invention;

FIG. 14 illustrates a thorax phantom tested in accordance with aspectsof the invention; and

FIG. 15 is a graph of the energy spectra of the simulated 80 kV X-raysource after being transmitted through certain K-edge filters accordingto aspects of the invention.

DETAILED DESCRIPTION OF THE INVENTION

Aspects of the invention are directed to systems and methods generatingspectral computed tomography data using pixelated K-edge coded aperturestructures. Advantageously, embodiments of the invention may overcomemany of the limitations of conventional methods used for SCT. Inaddition, by using pixelated K-edge coded aperture structures,lower-dose structured X-ray bundles that interrogate specific pixels ofthe target may be created. In one embodiment, advantageously, structuredillumination may be used to reduce the number of angles by sourcemultiplexing in limited angle geometries.

A pixelated K-edge coded aperture structure as referred to herein is astructure that has a filtering aspect and a pixelated coding aspect. Inone embodiment, the filtering aspect and the pixelated coding aspect areperformed by a single structure including a plurality of apertureshaving at least one K-edge filter incorporated therein. In anotherembodiment, the pixelated K-edge coded aperture structure includes afirst structure (e.g., patterned structure) for pixelating the X-raybeam(s) and a second structure (e.g., K-edge filter structure) forfiltering of the X-ray beam(s) that is separate from the firststructure.

FIG. 1 depicts a spectral CT imaging system 100 employing pixelatedk-edge apertures for generating spectral computed tomography data inaccordance with aspects of the invention. As a general overview,spectral CT imaging system 100 includes an X-ray generator 110, an X-raybeam detector 120, and a pixelated k-edge aperture structure 130.

X-ray beam generator 110 is configured for producing a plurality ofX-ray beams 112 for imaging a target. Suitable X-ray beam generators 110include those configured to produce monochromatic and/or polychromaticX-ray beams 112. X-ray beam generator 110 may be configured to produceand/or deliver X-ray beams 112 in the form of cone beams, fan beams, orany other suitable forms for producing and/or delivering X-ray beams112. X-ray beam generator 110 may be coupled to a gantry configured forpositioning X-ray beam generator 110 relative to the target and/or X-raybeam detector 120. One of ordinary skill in the art would readilyrecognize suitable gantries for positioning X-ray beam generator 110relative to the target and/or X-ray beam detector 120. Additionallyand/or alternatively, the target may reside on a patient positioningsystem that is configured to position and/or rotate the patient relativeto the X-ray beam generator 110 and/or X-ray beam detector 120. Suitablepatient positioning systems would be known to one of ordinary skill inthe art.

X-ray beam detector 120 is configured to detect and/or sense X-ray beams112 produced by X-ray beam generator 110. X-ray beam detector 120,preferably, has a geometry corresponding to X-ray beams 112 generator byx-ray beam generator 110. For example, X-ray beam detector 120 may be aline detector that is adapted for detecting a X-ray fan beam 112produced by X-ray beam generator 110. In another example, X-ray beamdetector 120 may be a two dimensional detector that is adapted fordetecting an X-ray cone beam 112 produced by an X-ray beam generator120.

As mentioned above, pixelated K-edge coded aperture structure 130 is astructure that has a filtering aspect and a pixelating aspect. Thepixelating aspect and the filtering aspect may be performed by a singlestructure or by two or more structures. As shown in FIG. 2A, pixelatedK-edge coded aperture structure 130 may be a single structure 132delineating a plurality of openings 134 (e.g., apertures) associatedwith at least two K-edge filters, where any individual opening of theplurality of opening 134 may contain a single filtering material. Forexample, each of the plurality of openings 134 may contain at least oneK-edge filter incorporated therein. A pair of balanced K-edge filtersmay be formed by a pair of K-edge filters contained within acorresponding pair of the plurality of openings 134 delineated by singlestructure 132, i.e., pixelated K-edge coded aperture structure 130.

Additionally and/or alternatively, each of the openings 134 of thepixelated K-edge coded aperture structure 130 is arranged with respectto the other openings 134 to spatially encode the plurality of X-raybeams transmitted through the plurality of openings 134 of pixelatedK-edge coded aperture structure 130. For example, each of the pluralityof openings 134 may be arranged to form a pattern, such that X-ray beams112 transmitted through the pixelated K-edge coded aperture structure130 are encoded to form a pattern. In one embodiment, the plurality ofopenings 134 are arranged in a randomized pattern, such that the X-raybeams 112 transmitted through pixelated K-edge coded aperture structure130 are encoded with a randomized pattern. In another embodiment,plurality of openings 134 are arranged in a non-randomized pattern, suchthat the X-ray beams 112 transmitted through pixelated K-edge codedaperture structure 130 are encoded with a non-randomized pattern thatmay be optimized to improve spectral CT imaging.

In another embodiment, pixelated K-edge coded aperture structure 130includes a first structure 136 (e.g., patterned structure) forpixelating the plurality of X-ray beams 112 and a second structure 140(e.g., K-edge filter structure) for filtering of the plurality of X-raybeams that is separate from the first structure 136. As illustrated bythe embodiment shown in FIG. 2D, first structure 136 for pixelatingX-ray beams 112 may be a block/unblock structure that includes aplurality of openings 138 (e.g., apertures). The block/unblock firststructure 136 may be formed of materials, such that X-ray beams 112contacting the block/unblock first structure 136 are blocked and/orprevented from passing through block/unblock first structure 136 whileX-ray beams 112 transmitted through the plurality of openings 138 arepermitted to passing through block/unblock first structure 136. Asmentioned above, the plurality of openings 138 may be arranged in apattern such that X-ray beams 112 transmitted through pixelated K-edgecoded aperture structure 130 are encoded to form a pattern. Theplurality of openings 138 may be arranged in a randomized pattern toencode the X-ray beams 112 transmitted through the pixelated K-edgecoded aperture structure 130 with a randomized pattern or arranged in anon-randomized pattern to encode the X-ray beams 112 transmitted throughpixelated K-edge coded aperture structure 130 with non-randomizedpattern. Although FIG. 2D illustrates a single K-edge filter for secondstructure 140, a different K-edge filter may be used for secondstructure 140 in another scanning position.

Pixelated K-edge aperture structure 130 is configured to filter theplurality of X-ray beams 112 using at least two K-edge filters that format least one pair of K-edge filters. Preferably, the pair of K-edgefilters are a balanced pair of K-edge filters. Additionally and/oralternatively, pixelated K-edge aperture structure 130 is configuredsuch that the difference of the spectra of the plurality of X-ray beamstransmitted through the plurality of openings has an energy bandcorresponding to a difference between K-edge values of a correspondingbalanced pair of K-edge filters. In one embodiment, at least one pair ofK-edge filters are formed of two materials having a difference in atomicnumber is 16 or less. In another embodiment, at least one pair of K-edgefilters are formed of two materials having a difference in atomic numberis 16 or more. The K-edge filters may form a Ross filter.

More generally, a K-edge filter is a material consisting of a high-Zelement, which is an element with a high atomic number, such astantalum, tungsten, or molybdenum, which sharply cuts off part of theX-ray spectrum above the element's K-shell electrons' binding energy. Byusing different K-edge filters in sequential scans, mono-energetic X-rayflux measurements can be obtained and used to reconstruct the linearattenuation coefficients of a particular energy bin. This schemeovercomes the limitations of photon counting detectors. By subtractingthe X-ray spectrum acquired from X-ray beams(s) transmitted through thefirst K-edge filter from the spectrum acquired from X-ray beams(s)transmitted through the second K-edge filter of the pair of K-edgefilters, a quasi-monochromatic curve may be obtained at the energy binbetween the K-edges of the pair of K-edge filters.

FIG. 3 depicts a method 300 of generating spectral computed tomographydata in accordance with aspects of the invention. As a general overview,method 300 includes generating a plurality of X-ray beams; encoding theplurality of X-ray beams by transmitting the plurality of beams througha pixelated K-edge coded aperture structure; detecting the encodedplurality of X-ray beams; and reconstructing a spectral CT image fromthe encoded plurality of X-ray beams.

In step 310, a plurality of X-ray beams 112 are generated, e.g., usingX-ray generators 110 discussed herein. The target may be scanned withthe plurality of X-ray beams 112 during at least one pass of an X-raybeam generator with respect to a target. In one embodiment, the targetis scanned during a first pass of the X-ray generator 110 relative tothe target and scanned during a second pass of the X-ray generator 110relative to the target. In another embodiment, however, the target isscanned solely during a single pass of the X-ray generator 110 relativeto the target. The plurality of X-ray beams 112 may be generated duringstep 310 may be monochromatic and/or polychromatic X-ray beams 112.Additionally, the plurality of X-ray beams 112 may be delivered to thetarget in the form of cone beams, fan beams, parallel beams, or anyother suitable form for producing and/or delivering X-ray beams 112.

In step 320, the plurality of X-ray beams 112 are encoded bytransmitting the plurality of beams through a pixelated K-edge codedaperture structure 130. The plurality of X-ray beams 112 may form arandom or non-random pattern after being encoded by way of transmissionof the plurality of X-ray beams 112 through the pixelated K-edge codedaperture structure 130. For example, in an embodiment utilizing at leasttwo passes, a first set of X-ray beams 112 are encoded during a firstpass of the X-ray generator 110 relative to the target as the first setof X-ray beams 112 is transmitted through pixelated K-edge aperturestructure 130 and a second set of X-ray beams 112 are encoded during asecond pass of the X-ray generator 110 relative to the target as thesecond set of X-ray beams 112 is transmitted through pixelated K-edgeaperture structure 130. In the embodiment utilizing solely a singlepass, the plurality of X-ray beams 112 are encoded during the singlepass of the X-ray generator 110 relative to the target.

In step 330, the encoded plurality of X-ray beams 112 are detected,e.g., using X-ray beam detector 120 discussed herein. X-ray beamdetector 120 may detect an intensity of the encoded plurality of X-raybeams 112. For example, X-ray beam detector 120 may detect a firstintensity of encoded X-ray beams 112 transmitted through a first K-edgefilter and detect a second intensity of encoded X-ray beams 112transmitted through a second K-edge filter. Although not illustrated,method 300 may include the step of determining the quasi-monochromaticintensities associated with the encoded X-ray beams 112, e.g., bysubtracting the first intensity of the encoded X-ray beams 112associated with a first K-edge filter from a second intensity of theencoded X-ray beams associated with a second K-edge filter. Thedetermination of the quasi-monochromatic intensities by the subtractionof the first intensity of the encoded X-ray beams 112 associated with afirst K-edge filter from a second intensity of the encoded X-ray beamsassociated with a second K-edge filter is just one example, otherprocedures may be employed to determine the quasi-monochromaticintensities associated with the encoded X-ray beams 112.

In step 340, a spectral CT image is reconstructed from the encodedplurality of X-ray beams 112. Reconstruction algorithms, such as thosediscussed herein, may be employed to facilitate reconstruction of thespectral CT image from the detected plurality of X-ray beams 112. In oneembodiment, the spectral CT image of the target is reconstructed usingquasi-monochromatic intensity measurements, e.g., by solving an inverseproblem.

FIG. 4 is a schematic depicting non-limiting, exemplary steps forobtaining spectral CT images. The detected encoded plurality of X-raybeams 112 may be used to obtain the quasi-monochromatic measurements bysolving an inverse problem. For example, low-rank minimization may beused to obtain complete filtered sinograms, which may be subtracted bypairs as in the embodiment utilizing multiple shots of X-ray beams 112and/or two or more passes of X-ray generator 110 relative to the target.By way of another example, sparsity promoting algorithms may be used toreconstruct CT images from the detected encoded plurality of X-ray beams112 and, subsequently, using these images to obtain the completefiltered sinograms. These sinograms are then used to obtain thequasi-monochromatic measurements by solving a least squares problem;subtracting the complete measurements is another alternative to obtainthe quasi-monochromatic measurements. In both of the above procedures,conventional CT algorithms, algebraic reconstruction methods or sparsitypromoting algorithms can be used to reconstruct the spectral CT images.

FIG. 5 is a schematic depicting further non-limiting, exemplary stepsfor obtaining spectral CT images. In order to filter the plurality ofX-ray beams 112, a K-edge filter from a balanced pair may be used ineach projection angle and a block/unblock coded pattern first structure136 may be positioned before or after the second filter structure 140.Thus, the obtained measurements in each angle may be obtained having asingle K-edge filtered X-ray projection with an illumination patternprovided by the block/unblock coded pattern first structure 136.Sinograms may be used to obtain the quasi-monochromatic measurementsand, ultimately, reconstruct the spectral CT images as further discussedherein.

FIG. 6 is a schematic depicting additional non-limiting, exemplary stepsfor obtaining spectral CT images. As illustrated by the embodiment shownin FIG. 6, spectral imaging systems 100 may include two or more X-raygenerators 110 and a second structure 140 comprising two separate K-edgefilters that together form a balanced pair. Each of the K-edge filtersmay be associated with a corresponding X-ray generator 110 and ablock/unblock coded second structure 136 may be positioned before orafter each of the K-edge filter. These block/unblock coded secondstructure 136 are, preferably, configured to be complementary with eachother. In other words, the coded aperture elements of block/unblockcoded second structure 136 associated with a particular detector pixelare such that only one of them has an unblocking element, while theothers correspond to a blocking element. For example, as illustrated inFIG. 6, if the coded aperture of Source 1 has an unblocking element(e.g., an opening or aperture) in a particular position, the codedaperture of the Source 2 has a blocking element in such position.

FIGS. 7-10E illustrate non-limiting embodiments of spectral CT imagingsystems utilizing different combinations of X-ray beam generators, X-raybeam detectors, and pixelated K-edge aperture structures.

FIG. 7 illustrates a non-limiting embodiment of a spectral CT imagingsystem 700 that includes an X-ray beam generator 710 that produces X-rayfan beams 712 and an X-ray detector 720. X-ray beam generator 710 may beconfigured to include collimator or be configured such that plurality ofgenerated X-ray beams 712 are fan beams, as shown in the FIG. 7. X-raydetector 720 is configured as a line detector in the illustratedembodiment. In this embodiment, to obtain a reconstruction of thetwo-dimensional slice being imaged, X-ray beam generator 710 and X-raydetector 720 rotate around the target to obtain projections at differentangles. Pixelated K-edge aperture structure 130 may include differentK-edge filters that are chosen from a set of balanced K-edge filters. Invarious embodiments, the selection of the set of balanced K-edge filtersare random, uniform, or the same for all the positions but different forparticular X-ray fan beams 712. Although pixelated K-edge aperturestructure 130 is illustrated in FIG. 7 as being positioned between X-raybeam generator 710 and the target, pixelated K-edge aperture structure130 may be positioned in front of or behind the target so long aspixelated K-edge aperture structure 130 is positioned between X-ray beamgenerator 710 and X-ray detector 720. Spectral CT imaging system 700 mayalso obtain projections by rotating the target while the position ofX-ray generator 710 and X-ray detector 720 are stationary. Theprojections can be from limited angles in a limited range of thepossible circular trajectory or they can be taken at multiple positionsin the full 360° range.

FIG. 8 illustrates a non-limiting embodiment of a spectral CT imagingsystem 800 that includes an X-ray beam generator 810 that produces X-raycone beams 812 and an X-ray detector 820. X-ray generator 810 may beconfigured to have a collimator or be configured such that the X-rayillumination is a X-ray cone beam 812, while X-ray detector may be a twodimensional detector, as shown in FIG. 8. X-ray generator 810 and X-raydetector 820 may rotate around the target to obtain projections atdifferent angles. Pixelated K-edge aperture structure 130 have differentK-edge filters that are chosen from a set of balanced filters, asdiscussed herein. Although pixelated K-edge aperture structure 130 isillustrated in FIG. 8 as being positioned between X-ray beam generator810 and the target, pixelated K-edge aperture structure 130 may bepositioned in front of or behind the target so long as pixelated K-edgeaperture structure 130 is positioned between X-ray beam generator 810and X-ray detector 820. Spectral CT imaging system 800 may also obtainprojections by rotating the target while the position of X-ray generator810 and X-ray detector 820 are stationary. The projections can be fromlimited angles in a limited range of the possible circular trajectory orthey can be taken at multiple positions in the full 360° range.

FIGS. 9A and 9B illustrate non-limiting embodiments of a spectral CTimaging system 900 that includes an X-ray beam generator 910 thatproduces X-ray beams 912 and an X-ray detector 920. Spectral CT imagingsystem 900 as depicted in FIG. 9A includes a first X-ray generator 910A,a second X-ray generator 910B, a first pixelated K-edge aperturestructure 130A, a second pixelated K-edge aperture structure 130B, afirst X-ray detector 920A, and a second X-ray detector 920B. In oneembodiment, the n^(th) position for pixelated K-edge aperture structure130A has the pair filter of the material contained in the n^(th)position for pixelated K-edge aperture structure 130B, such that themeasurements can be subtracted to obtain a monochromatic reading.Although X-ray detectors 920A and 920B are positioned horizontally withrespect to the conveyer belt in FIG. 9A, in another embodiment X-raydetectors 920A and 920B may be positioned substantially parallel or atan angle with respect to the conveyer belt. Spectral CT imaging system900, as depicted in FIG. 9B, includes an X-ray generator 910 thatproduces X-ray beams 912 in the form of X-ray cone beams and a twodimensional X-ray detector 920.

FIGS. 10A-10E illustrate non-limiting embodiments of a spectral CTimaging system 1000 that includes an X-ray beam generator 1010 thatproduces X-ray beams 1012 and an X-ray detector 1020. Spectral CTimaging system 1000 includes features that are employed similar to thosediscussed above with respect to spectral CT imaging systems 100 and700-900, with details regarding those features omitted in order to avoidduplication.

X-ray beam generator 1010 is illustrated in FIGS. 10-10E as producing aplurality of X-ray beams 1012 in the form of X-ray fan beams or X-raycone beams, while X-ray beam detector 120 is illustrated as a line X-raydetector or a two dimensional X-ray detector. In FIGS. 10A-10E,pixelated K-edge coded aperture structure 130 is illustrated asincluding a first structure 136 (e.g., a block/unblock patternedstructure) for pixelating the X-ray beams 1012 and a separate, secondstructure 140 comprising two separate K-edge filters for filtering X-raybeams 1012. A K-edge filter may be positioned between each of the X-raygenerators 1010 and X-ray detector(s) 120. Under the circumstances wherethe target rotates and X-ray generator 1010 and X-ray detector 120remain stationary, a different K-edge filter may be positioned inbetween of X-ray generator 1010 and X-ray detector 120 for eachmeasurement taken. In an alternative embodiment, projections may beobtained by rotating the target while the position of X-ray generator1010 and X-ray detector 1020 are stationary. The projections can be fromlimited angles in a limited range of the possible circular trajectory orthey can be taken at multiple positions in the full 360° range. Inembodiments of spectral CT imaging system 1000 having an X-ray generator1010 that produces X-ray cone-beam, X-ray detector 102 and the targetmay be configured to remain stationary while X-ray generator 1010rotates on an arch or moves in a linear trajectory over the target. Inthese embodiments, each projection is preferably paired with a differentK-edge filter and a different first structure 136 having a plurality ofopening 138 forming a different coded pattern for the X-ray beams 1012transmitted therethrough. Additionally and/or alternatively, multipleX-ray generators 1010 may be available to obtain sequential projectionsat different positions (instead of rotating the same X-ray generators101).

EXAMPLES

Non-limiting examples are described below to illustrate variousimplementations and simulated experiments of embodiments of theinvention and/or to elucidate advantages associated with variousembodiments of the invention.

Example 1

Simulated experiments were performed using conventional X-ray imagingsystems with K-edge coded aperture masks to obtain spectrallymultiplexed measurements used to reconstruct the energy-binned images.In this Example, filter pairs were aligned with each X-ray beam in amulti-shot architecture—therefore obtaining compressive measurements inboth the spectral and spatial domains. This approach may be referred toherein as compressive spectral X-ray imaging or CSXI.

Quasi-monochromatic X-rays were be obtained using the balanced filter bysubtracting the X-ray spectrum acquired with one filter from thespectrum acquired with the second filter of the pair. Specifically, thequasi-monochromatic curve was obtained at the energy bin between theK-edges of the filters. For the process used in this Example, two ormore shots could be used in which the pixels of the coded apertures arechosen from a set of balanced Ross filters, such that a particular pairis assigned to a particular detector position.

The alternating direction method of multipliers (hereafter “ADMM”) wasused to solve the highly ill-posed problem by exploiting the inherentsparsity of X-ray images in the spatial domain and the low-rankstructure of the data-cube. Although not completed in this Example, suchprocessed image data may be further decomposed into basis functions toobtain material based images that are useful for diagnosis and analysisin medical and/or security applications.

For a polychromatic X-ray beam passing through an target, the X-rayattenuation at each energy level, E, was obtained from the polychromaticBeer-Lambert law asI(E)=S(E)·Q(E)·P(E)exp(−∫_(l)μ(l,E)dl)  Equation 1

where μ(l, E) is the linear attenuation coefficient of the target at theposition l; S(E) is the X-ray source spectrum; Q(E) is theenergy-dependent detector response; and P(E) is the photon energy at theenergy E. For photon counting detectors P(E)=1 and for conventionalintegrating detectors P(E)=E. That is, photon counting detectors reflectthe number of photon counts in each energy bin. Integrating detectors,on the other hand, accumulate the intensity over the entire X-ray energyspectrum; thus, the contribution of each photon to the reading isweighted by its original energy. Let I₀(E)=S(E)·Q(E)·P (E) be theintensity measured at energy E without any target in front of the X-raysource; then, for a conventional X-ray imaging system, with integratingdetectors, the intensity registered by the j^(th) detector element isgiven by:I _(j)=∫_(E) I ₀(E)exp(−∫_(l)μ(l,E)dl)dE.  Equation 2

If P is the total number of X-ray source positions and M is the numberof detector elements in the detector array, then j=1, . . . , MP. Theintegrated grayscale data cannot be used to reconstruct directly theenergy-binned images μ(l, E), as it does not provide spectrally resolveddata. To obtain quasi-monochromatic X-ray spectra Ross filter pairs wereused. Ross filters consist of materials with nearly adjacent atomicnumbers whose thicknesses are carefully matched such that thetransmitted spectra are identical for all photon energies except in thenarrow energy bin between their respective K-edges. Thequasi-monochromatic measurements were then obtained by subtracting theX-ray intensity acquired using the filter with the lower K-edge fromthat with the higher K-edge in the Ross pair. Therefore, the bandwidthof the energy bin is defined by the difference between the K-edgeenergies of the two elements constituting the Ross pair.

FIG. 11 illustrates the process to obtain a quasi-monochromatic X-rayspectrum using the Ross pair constituted by Dysprosium (Dy) and Cerium(Ce). The thickness of each filter was set such that the X-raytransmission curves of the filtered spectra were nearly the same forboth filters over the entire energy spectrum except within the narrowband between the K-edges of the elements. As shown in FIG. 11, bysubtracting the spectrum filtered by Ce from the spectrum filtered by Dya quasi-monochromatic spectrum between 40.4 keV and 53.8 keV wasobtained.

To calculate the intensity of the X-ray beam at the energy E afterpassing through a homogeneous filter, the following equation was used.I _(f)(E)=I ₀(E)exp[−μ_(f)(E)δ_(j,f)],  Equation 3

where μf (E) is the linear attenuation coefficient of the filter f atthe energy E, and δ_(j,f) is the length of the intersection of thej^(th) X-ray beam with the filter f. The latter is given byδ_(j,f)=ρ_(f)/cos(ψ_(i)), where ρ_(f) is the thickness of the filter andis the angle between the normal of the filter and the j^(th) X-ray beam.Thus, the filtered measurements on the j^(th) element using anintegrating detector are given by:I _(j) ^(f)=∫_(E) I _(f)(E)exp[−∫_(l)μ(l,E)dl]dE  Equation 4

CSXI process here in this Example included only two scans to obtain thelinear attenuation coefficient at all the energy bins; therefore, thescanning time and the radiation dose were reduced without compromisingthe image quality which is of paramount importance in medical imaging.CSXI process also used K-edge coded apertures as multiple materialfilters to obtain spatially and spectrally coded illuminationprojections of the target. For simplicity, in the simulated experimentsof this Example, the coded apertures were assumed to have the samenumber of elements as the detector array and the coded aperture pitchwas fixed to obtain one to one correspondence with the detectorelements. More general scenarios where there is no one to onecorrespondence with the detector array would have the same forwardmodel; however, computational methods to treat the pixel mismatch wouldhave to be implemented such as the approach used for coded aperturesnapshot spectral imaging (CASSI) systems.

Different coded aperture patterns were used for each view angle positions_(p), where p=1, . . . , P, and each shot t where t=1, . . . , T asshown in FIG. 12, which depicts the CSXI system for a fan beamarchitecture with T=2 shots. It should be noted that the measurementsobtained by a single scan were not sufficient for reconstruction sincethe quasi-monochromatic intensity was obtained by subtracting the X-rayintensities of a Ross filter pair. Thus, at least 2 scans were performedin which a Ross pair was assigned to each detector position j. In thisway, in the first shot t=1, the K-edge coded aperture contained one ofthe materials of the pair in the position j and in the second shot, t=2,the coded aperture contained the other material from the pair in thesame position, as shown in FIG. 12. Using these measurements, thequasi-monochromatic measurement for the k^(th) energy bin at the j^(th)detector element could be obtained as:d _(j) ^(k) =I _(j) ^(f) ¹ ^(k) −I _(j) ^(f) ² ^(k)   Equation 5

where I_(j) ^(f) ¹ ^(k) and I_(j) ^(f) ² ^(k) are the measurement takenby the filters with higher and lower k-edges in the Ross pair associatedwith the k^(th) energy bin, respectively. FIG. 2 depicts themeasurements obtained using Dy and Ce, I^(f) ¹ ^(k) =[I^(f) ¹ ^(k) , . .. , where I_(MP) ^(f) ¹ ^(k) ]^(T) and, I^(f) ² ^(k) =[I₁ ^(f) ² ^(k) ,. . . , where I_(MP) ^(f) ² ^(k) ]^(T) respectively. The discretized setof measurements, referred to as the sinogram, corresponds in each caseto the measurements of the detectors paired with the respective filter.Note both sinograms contain information in the same location such thatthe filtered measurements can be subtracted to obtain thequasi-monochromatic sinogram. The measurements obtained from Equation 5can be considered mono-energetic, and thus they can be described by theBeer-Lambert Law as

d_(j)^(k) = d_(j 0)^(k)  exp [−∫_(ℓ)μ(ℓ, k)d ℓ], where d_(j0) ^(K) is the quasi-monochromatic intensity obtained usingthe Ross filter pair when there is no target in front of the X-raysource, and μ (l, k) is the linear attenuation coefficient at the k^(th)energy bin. Consequently, the log-transformed measurements at the k^(th)energy bin can be obtained as:

$\begin{matrix}{y_{j}^{k} = {\ln\frac{d_{j\; 0}^{k}}{d_{j}^{k}}}} & {{Equation}\mspace{14mu} 6}\end{matrix}$

Equation 6 only applies to the kth energy bin corresponding to the Rossfilter pair assigned to the position j, for any other energy bins y_(j)^(m)=0, whenever m does not equal k. For example, the Ross pair Dy-Cecorresponds to the second energy bin, k=2, then for all j positionswhere Dy-Ce is assigned

$y_{j}^{k} = {\ln\frac{d_{j\; 0}^{k}}{d_{j}^{k}}}$for K=2, and y_(j) ^(k)=0, where k does not equal 2. As a result, theenergy-binned sinograms are sparsely distributed as shown in FIG. 12.Five different filter materials [Molybdenum (Mo), Cerium (Ce),Dysprosium (Dy), Erbium (Er) and Tungsten (W)] were used for the K-edgecoded apertures. The Ross pairs, their thicknesses (ρ) and thecorresponding energy bins for these materials are shown in Table 1,provided on the following page. Discretizing the line integrals and thelinear attenuation coefficient μ(l, k), the energy-binned measurementsy^(k)=[y₁ ^(k), y₁ ^(k), . . . y_(MP) ^(k)]^(T) are given by thefollowing linear model:y ^(k) =C ^(k) Hx ^(k)  Equation 7

where x^(k)∈R^(N) ² ^(×1) is the vectorized linear attenuationcoefficient μ(l, k) of the N×N discretized target under H∈R^(MP×N) ² isthe CT system matrix such that the weights H_(j,i) account for thehardware settings, that is, H_(j,i) corresponds to the intersection ofthe j^(th) ray with the i^(th) pixel in the target, and C^(k) is thecoded aperture matrix. Each row of the matrix, H, corresponds to aparticular detector element j, and each detector is associated with aparticular Ross filter pair which in turn is associated with aparticular energy bin k. Thus, the elements of the coded aperture matrixC^(k) select the rows of the matrix H associated with the Ross filterpair that corresponds to the kth energy bin. Mathematically, the codedaperture matrix C^(k) is defined as a diagonal binary matrix, where[C^(k)]_(j,l)=0 for 3 does not equal l and [C^(k)]_(j,j)=1 if the K-edgecoded aperture element associated with the j^(th) detector contains theelements of the Ross filter pair corresponding to the k^(th) energy bin,otherwise [C^(k)]_(j,j)=0.

TABLE 1 k Filter 1 Filter 2 Energy (keV) ρ₁ (μm) ρ₂ (μm) 1 Ce Mo20.0-40.4 52.8 74.7 2 Dy Ce 40.4-53.8 30.6 52.8 3 Er Dy 53.8-57.5 26.730.6 4 W Er 57.5-69.5 9.9 26.7

FIG. 13 shows the coded aperture matrices for a system with M=6, P=1,T=2, and the filter pairs in Table 1. The first energy bin was definedby the pair Ce-Mo, for the example in FIG. 13, this filter pair wasassigned to j=3 and j=4, thus the only non-zero elements in the codedaperture matrix were [C¹]_(3,3)=[C¹]_(4,4)=1. The coded apertures forthe remaining pairs were also depicted for the source position s₁.

The implementation of coded aperture masks with a pixel pitch having aone to one correspondence with the pixels on the detector can be achallenge due to the magnification factor when the codes are placedclose to the source and before the target under inspection. Ageneralized model when a coded aperture element impinges onto multipledetector elements could be developed. Another approach is to model thesystem in which the coded aperture mask is located after the target andclose to the detector. If the coded aperture elements have a one to onecorrespondence with the pixels on the detector, the measurements from anintegrating detector when the mask is placed on the detector side aregiven by:

$\begin{matrix}\begin{matrix}{I_{j}^{f} = {\int_{E}{{I_{0}(E)}\exp\{ {- \lbrack \ {{\int_{\ell}{{\mu( {\ell,E} )}d\;\ell}} + {{\mu_{f}(E)}\delta_{j,f}}} \rbrack} \}{dE}}}} \\{= {\int_{E}{{I_{0}(E)}{\exp\lbrack {{- {\mu_{f}(E)}}\delta_{j,f}} \rbrack}{\exp\lbrack {- {\int_{\ell}{{\mu( {\ell,E} )}d\;\ell}}} \rbrack}{dE}}}}\end{matrix} & {{Equation}\mspace{14mu} 8}\end{matrix}$

where μ_(f)(E) is the linear attenuation coefficient of the filter f atthe energy E, and, δ_(j,f) is the length of the intersection of thej^(th) X-ray beam with the filter. Note, equation 8 is equivalent to theresult obtained in Equation 4. Thus, the formulation is also valid whenthe coded aperture is placed on the detector side.

Ross filter pairs were assigned randomly to each detector element jfollowing a uniform distribution; as a result, the number ofmeasurements per energy bin is given by D=MP/K, where K is the totalnumber of energy bins. The reconstruction of each energy bin is thus ahighly ill-posed problem. Hence, to effectively recover the data cubefrom the compressed measurements, regularization constraints were addedbased on the structure of the data. In this Example, a joint sparse andlow-rank optimization method was used. This approach seeks to jointlyminimize the l₁ norm of the sparse representation of the data cube, andits nuclear norm. Furthermore, the tensor modeling was used to formulatethe reconstruction problem. The following concepts are used in theremaining of the paper:

A. Tensor Modeling

1) Vectorization: Denoted by vec(χ), where vec:R^(Q1×Q2× . . . ×QN)→R^(Q1·Q2 . . . QN) is an operator that stacks theentries of a tensor in reverse lexicographical order into a Q₁ Q₂ . . .Q_(N)-long column vector [24].

2) Unfolding: For a tensor χ, the matrix unfolding denoted byχ_((n))=Unfold n-th (χ) is defined by the mapping of:R^(Q1×Q2× . . . ×QN)→R^(Qn×(Qn+1·Qn+2 . . . QN·Q1·Q2 . . . Qn−1)). Theresulting matrix χ_((n)) is a 2D matrix with Q_(n) rows andQ_(n+1)·Q_(n+2) . . . Q_(N)·Q₁·Q₂ . . . Q_(n−1) columns.

3) Folding: It is the inverse operator of unfolding and it is defined byχ=Fold n-th(χ_((n))) [8].

Using these concepts, the fan beam model used in this Example can begeneralized to multiple geometries by representing the spectral linearattenuation coefficients with a tensor X∈R^(Q1×Q2× . . . × Qξ+1), whereξ is the dimensionality of a single energy X-ray image. For instance,ξ=2 for fan-beam and parallel beam CT, ξ=3 for cone-beam CT and ξ=4 fordynamic cone beam CT. Without loss of generality, the remaining of thisExample treats the inverse problem for a fan-beam system; therefore, thetensor X is a multi-dimensional matrix∈R^(N×N×K), where the first twodimensions are spatial, the third is the energy dimension, andvec(X)=[x^(1 T)|x^(2 T)| . . . |x^(K T)]^(T). The spectral CT model in(7) can thus be generalized as:

y = A(x) = [[A¹x¹]^(T)…[A^(k)x^(k)]^(T)]^(T)where γ is the vertical concatenation of the quasi-monochromaticsinograms γ^(k) A^(k)=C^(k)H is the sensing matrix for the k^(th) energybin and A (*) is the tensor expression used to generalize the forwardprojection above. As it can be seen, the reconstruction of X from themeasurements y describes an ill-posed problem that cannot be solvedusing traditional LS approaches. Nonetheless, the compressive sensingprinciples can be used to recover the data-cube without loss ofreconstruction fidelity. This framework can be used as long as vec(X) issufficiently sparse in some basis ψ and such basis is incoherent withthe forward measurement matrix. Let X be represented byvec(X)=ψ[vec(θ)], whereθ is the sparse tensor representation of the object, and ψ∈R^(N) ²^(K×N) ² ^(K) is the representation basis. Then, the sparserepresentation of the object x can be reconstructed by solving thefollowing optimization problem:{circumflex over (θ)}=argmin ½∥y−A(X)∥₂ ²+λ₁∥vec({circumflex over(θ)})∥₁B. CSXI Reconstruction via ADMM

An additional regularization constraint is added to the inverse problem,above, to take into account the correlation across spectral channels.This is achieved by introducing a low-rank constraint on χ, such that:{circumflex over (θ)}=argmin ½∥y−A(X)∥₂ ²+η₁∥vec({circumflex over(θ)})∥₁+η_(*)∥χ∥*.The ADMM algorithm is adapted to solve immediately preceding equation asit provides the necessary tools to split the optimization problem intosmall convex optimization problems that can be solved using simpleralgorithms. When the augmented Lagrangian is calculated, twointermediate variables to separate the l₁, l₂ and nuclear normcomponents are introduced. Thus, immediately preceding equation istransformed as follows:

$( {X,D,B} ) = {{\underset{{X.D},B}{argmin}\frac{1}{2}{{y - {A(X)}}}_{2}^{2}} + {\lambda*{D}*{+ \lambda_{1}}{B}_{1}} + {\mu*{{D - X - V}}_{2}^{2}} + {\mu_{1}{{B - {\Psi^{- 1}(X)} - W}}_{2}^{2}}}$The problem is then split into three different sub-problems summarizedin Algorithm 1. Step 2 of the algorithm is an l₂ minimization problemwhich was solved using the conjugate gradient (CG) algorithm with afixed step size, and the solution for Step 3 and Step 4 can be found as:

${D^{\tau + 1} = {\frac{1}{\xi + 1}{\sum\limits_{\eta = 1}^{\xi + 1}\;{{fold}( {S_{\xi}\{ {X_{(n)}^{\tau + 1} + V_{(n)}^{\tau}} \}} )}}}},{B^{\tau + 1} = {{softshrink}\{ {{{\Psi^{- 1}( X^{\tau + 1} )} + W^{\tau}},{\lambda_{1}/\mu_{1}}} \}}},$respectively; where ∈=λ*/μ*, S_(∈){⋅} is the shrinkage operator withparameter ∈, and τ is the iteration number. Note X_((n)) and V_((n)) arethe matrices obtained by applying the unfolding operator to the tensorsχ and V, respectively.

Algorithm 1 ADMM for CSXI reconstruction Input: D₀ = X₀, B₀ = ψ⁻¹(X₀)and V₀ = W₀ = 0.  1: for τ = 0 to max_(iter) − 1 do  2:   𝒳 τ + 1 = arg ⁢⁢min 𝒳 ⁢ ⁢ 1 2 ⁢  𝓎 - ⁢ ⁢ μ * ⁢  τ - 𝒳 τ - τ ⁢  2 2 ⁢ ⁢ + ⁢ μ 1 ⁢  τ - ⁢ ⁢ Ψ - 1 ⁡( 𝒳 τ ) - τ ⁢  2 2  3:   τ + 1 = argmin ⁢ ⁢ λ * ⁢  τ ⁢  * ⁢ ⁢ + ⁢ μ * ⁢  τ +𝒳 τ + 1 - τ ⁢  2 2  4:   τ + 1 = arg ⁢ ⁢ min ⁢ ⁢ λ 1 ⁢  τ ⁢  1 ⁢ ⁢ + ⁢ μ 1 ⁢ τ - Ψ - 1 ⁡ ( 𝒳 τ + 1 ) - τ ⁢  2 2  5:  

^( τ+1) =

 ^(τ) + χ^(τ+1) −

 ^(τ+1),

 ^(τ+1) =

 ^(τ) + Ψ⁻¹(χ^(τ+1)) −

 ^(τ+1)  6: end for  7: return χ^(τ+1)C. Initialization

The X-ray attenuation coefficients vary smoothly and have sharpboundaries. Total variation (TV) regularization is good reconstructionmethod for such cases, as it is robust to noise and the edges in thereconstructions are well-defined. Hence, in order to find a suitableinitialization point for the ADMM algorithm, the energy-binned imagesx^(k) are reconstructed from the corresponding quasi-monochromaticsinograms by solving the following minimization problem independentlyfor each energy bin:

${x^{\hat{k}} = {{\underset{x^{k}}{argmin}\mspace{14mu}{{y^{k} - {A^{k}X^{k}}}}_{2}^{2}} + {\lambda\;{{TV}( x^{k} )}}}},x^{k}$where λ is a regularization constant, ∥.∥₂ corresponds to the l norm andTV(x) is the total variation of x defined as Σ²√{square root over((Δ_(i) ^(h)x)²+(Δ_(i) ^(v)x)²)} where Δ_(i) ^(h) and Δ_(i) ^(v) are thehorizontal and vertical first-order local difference operatorsrespectively. In this Example, the inverse problem described immediatelyabove is solved independently for each energy bin using the two-stepiterative shrinkage/thresholding algorithm (TwIST) and the result wasused as the initial value for the ADMM algorithm detailed in Algorithm1.

Example 2

A simulated experiment for an X-ray fan beam system with K-edge codedapertures was performed using a 256×256 modified Forbild thorax phantomgenerated using the CONRAD software and the processes discussed inExample 1. The modified phantom consists of eight different materials tosimulate lung, heart, artery, bone, soft tissue, air, iodine, andmarrow. FIG. 14 shows the thorax phantom tested in this Example, withthe tissues and materials defined in Table 2, provided below.

TABLE 2 Number Material Density (g/cm³) 1 Air 0.00 2 Lung 0.26 3 AverageSoft Tissue 1.00 4 Heart (Blood) 1.06 5 ROI (0.9% Iodine + 99.1% Blood)1.09 6 Artery (Blood) 1.06 7 Bone 1.50 8 Marrow 0.98

The mass attenuation coefficient were obtained from the NationalInstitute of Standards and Technology (NIST) X-ray attenuationdatabases. The X-ray filtered spectra I_(f)(E) were simulated at 80 keVusing the Spektr software and energy weighted integrals over 1 keVspectral steps were obtained for each filtered measurement according toEquation 4 from Example 1. The projection data were simulated using theASTRA tomography toolbox for a regular fan-beam X-ray systemarchitecture with P=450 view angles and M=336 detector elements perangle. The X-ray fan beam covered a 43.52 cm diameter field of view andthe dimensions of the discretized target were 40 cm×40 cm. A copper (Cu)filter of 0.25 mm was placed in front of the X-ray source to filter theenergies lower than 20 kV and the thicknesses of the elements of theRoss filter pairs were matched precisely to obtain energy-binnedsinograms.

After matching the Ross filter pairs, the thickness of the Mo, Ce, Dy,Er, and W filters were set to be 74.7, 52.8, 30.6, 26.7 and 9.9 μmrespectively. The energy bins defined by these Ross filter pairs were20.0-40.4, 40.4-53.8, 53.8-57.5, and 57.5-69.5 keV, for k=1, 2, 3, 4respectively, as detailed in Table 1 from Example 1. The filter pairsassigned to each detector position j were chosen randomly from Ce-Mo[20.0-40.4], Dy-Ce [40.4-53.8], Er-Dy [40.4-53.8] and W-Er [57.5-69.5].That is if Er-Dy was assigned to the detector position j=1, then in thefirst scan, the K-edge coded aperture contained Er at the position j=1and in the second scan it contained Dy at the same position. FIG. 15depicts the energy spectra of the simulated 80 kV X-ray source seenthrough the five different filters. Additionally, the resultinglog-transformed measurements, y^(k), were determined and the sinogramimages were transformed using a gamma correction with γ=0.4 to improvethe visualization contrast.

To evaluate the performance of the CSXI system and the reconstructionalgorithm discussed in Example 1, comparative reconstructions wereproduced using the system disclosed in Y. Rakvongthai et al., “SpectralCT using multiple balanced K-edge filters,” IEEE TRANS. ON MEDICALIMAGING, vol. 34, no. 3, pp. 740-747, March 2015 (hereafter“Rakvongthai”), which incorporated herein for all purposes. The systemproposed in Rakvongthai will be referred to as 5 Shot K-edge filtering(hereafter “5SKF”). In order to compare both systems, the radiation dosewas set to be equivalent; that is, the number of measurements acquiredwith each filter was set to be the same in both systems. For M=386 andP=450 in the 2 shot CSXI system, the number of angles in the 5SKF systemwas set to P=113 for Molybdenum and Tungsten and P=225 for the rest ofthe filters. In the ADMM algorithm the energy-binned CT image wasrepresented on the sparse basis ψ=ψ_(DCT)⊗ψ_(W), where ⊗ is theKronecker product, ψ_(DCT) is the discrete cosine transform (DCT) basisin the energy domain, and ψ_(W) is the 2D Haar wavelet basis in thespatial domain.

LS was used to obtain the energy-binned sinograms for the 5SKF systemand then the energy-binned images are independently reconstructed usingthe 2D-FBP algorithm available in Matlab (ifanbeam function). Thereconstructions of both systems were compared to the linear attenuationimages, obtained from the NIST database, corresponding to the centralenergy in each energy bin, that is, 35 keV for k=1, 48 keV for k=2, 56keV for k=3, and 64 keV for k=4. S was used to obtain the energy-binnedsinograms for the 5SKF system and then the energy-binned images wereindependently reconstructed using the 2D-FBP algorithm available inMatlab (ifanbeam function). The reconstructions of both systems werecompared to the linear attenuation images, obtained from the NISTdatabase, corresponding to the central energy in each energy bin, thatis, 35 keV for k=1, 48 keV for k=2, 56 keV for k=3, and 64 keV for k=4

The linear attenuation coefficients at the four energy bins, for thefour pixels highlighted in FIG. 14 were determined for both the 5SKF and2 shot CSXI systems as well as the reference images. The 4 pixelscorrespond to materials simulating lung, soft tissue, blood, and bone asdetailed in Table 2. It should be noted that the 2 shot CSXI systemprovided a more accurate approximation of the spectral information ofthe target than the 5SKF system for all energy bins.

Additionally, reconstructions from both systems and the reference imagesat each energy bin were obtained. The peak signal to noise ratio (PSNR)was used to evaluate the reconstructions since it is suitable forcomparing restoration results. For a scenario with an image I and areconstruction R of size N×N it is defined as

${{PSNR} = {10{\log_{10}( \frac{{Max}_{l}^{2}}{MSE} )}}},$where is the maximum possible pixel value of the image I and MSE is themean squared error given by

${MSE} = {\frac{1}{N^{2}}{\sum\limits_{i = 0}^{N - 1}{\sum\limits_{j = 0}^{N - 1}{\lbrack {{I( {i,j} )} - {R( {i,j} )}} \rbrack^{2}.}}}}$It was observed that high-quality images at all energy bins wereobtained using the CSXI system and the proposed algorithm. On the otherhand, the results obtained using the 5SKF system presented numerousartifacts since FBP is not a suitable algorithm to solve the inverseproblem when the number of view angles is limited. Furthermore, detectorsub-sampling results in higher quality reconstructions compared to anglesub-sampling. Thus, the structure of the CSXI framework allowed forhigher reconstruction quality as the sub-sampling is performed in thedetectors instead of the view angles. Additionally, it was evident thatthe 5SKF system was not able to reconstruct all of the features in theoriginal image—in this case, the ribs of the phantom—whereas the CSXIsystem was able to accurately reconstruct all of the features in thephantom.

Some filter elements were used more than once in the Ross pairs. Forexample, for the list of elements in Table 1, Ce was used for both k=1and k=2, Dy was used for both k=2 and k=3, and Er is used for both k=3and k=4. Thus, for a system with T>2 shots, instead of selecting aparticular filter pair for each detector position j, a trio of elementscould be assigned, such that one of the elements of the trio was used inmore than one energy bin. This trio assignment can increase the samplingnumbers in the sub-sampled sinograms, thus retaining more spectralinformation for each energy bin and improving the reconstructionperformance especially when a large number of filters are used.

Additional experiments for a different phantom with more detailedfeatures were performed for multiple values of P to evaluate the systemat low-dose scenarios. For this experiment the target was 256×256, thenumber of detector elements per view was set to M=512, the source todetector distance and the source to target distance was set to 80 cm and40 cm respectively and the detector length was set to 41.3 cm.Projections using the 5SKF system and the CSXI system for T=2 and T=3shots were obtained. As in the previous scenario, the number of viewangles P was set accordingly in each simulation so that the radiationdose was equivalent. For T=3, the elements of the K-edge coded apertureswere chosen from Mo-Ce-Dy and Dy-Er-W for each detector position j,which resulted in energy-binned sinograms with 50% sub-sampling. In thissimulation, the reconstructions of the 5SKF system were performed usingTV regularization to exploit the sparsity of the target and counteractthe limited view angle problem. It should be noted that the average PSNRof the reconstructions obtained with the CSXI system was higher for allthe view angles for both T=2 and T=3 compared to the 5SKF system.Additionally, the radiation dose required by the CSXI system to obtainan average PSNR of 33.3 dB is approximately 30% of the radiation dosenecessary in the 5SKF system to obtain a similar performance, which isof significant interest in medical applications. It was observed thatthere were more artifacts in the reconstructions obtained using the 5SKFsystem compared to the CSXI system even when using sparsityregularization constraints for both reconstructions. Furthermore, thePSNR improvement is up to 7 dB in the third and fourth energy bins.

While preferred embodiments of the invention have been shown anddescribed herein, it will be understood that such embodiments areprovided by way of example only. Numerous variations, changes, andsubstitutions will occur to those skilled in the art without departingfrom the spirit of the invention. Accordingly, it is intended that theappended claims cover all such variations as fall within the spirit andscope of the invention.

What is claimed is:
 1. A method for generating spectral computedtomography data for spectral X-ray image reconstruction, the methodcomprising the steps of: generating a plurality of X-ray beams; encodingthe plurality of X-ray beams by transmitting the plurality of X-raybeams through a pixelated K-edge coded aperture structure; detecting theencoded plurality of X-ray beams; and reconstructing a spectral CT imagefrom the encoded plurality of X-ray beams.
 2. The method of claim 1,wherein the pixelated K-edge coded aperture structure delineates aplurality of openings, each of the plurality of openings containing atleast one K-edge filter.
 3. The method of claim 2, wherein a pair ofK-edge filters contained within a corresponding pair of the plurality ofopenings form a pair of balanced K-edge filters.
 4. The method of claim3, wherein the pair of K-edge filters are formed of two materials havinga difference in atomic numbers of 16 or less.
 5. The method of claim 1,comprising transmitting the plurality of X-ray beams through a firstblocking/unblocking structure for pixelating the plurality of X-raysbeam and a second structure comprising at least one K-edge filter forfiltering of the plurality of X-ray beams, wherein the first structureis separate from the second structure.
 6. The method of claim 1, furthercomprising categorizing the encoded plurality of X-ray beams into energybins.
 7. The method of claim 1, wherein the restructuring of thespectral CT image includes using a quasi-monochromatic intensity.
 8. Themethod of claim 1, wherein the encoding the plurality of X-ray beamscomprises performing a first pass and a second pass on a target, and therestructuring of the spectral CT image of the target includes using aquasi-monochromatic intensity.
 9. The method of claim 1, wherein theencoding of the plurality of X-ray beams comprises helically scanningthe target.
 10. The method of claim 1, further comprising rotating thetarget while maintaining a detector of the encoded plurality of X-raybeams and a generator of the plurality of X-ray beams stationary. 11.The method of claim 1, further comprising rotating a generator of theplurality of X-ray beams and rotating a detector of the encodedplurality of X-ray beams.
 12. The method of claim 1, comprisinggenerating the plurality of X-ray beams with a plurality of X-raygenerators, each X-ray generator associated with a respective one of aplurality of pixelated K-edge coded aperture structures for encoding arespected X-ray beam.
 13. A method for spectral X-ray tomography, themethod comprising the steps of: scanning a target with a plurality ofX-ray beams during at least one pass of an X-ray beam generator withrespect to the target; encoding the plurality of X-ray beams during theat least one pass by transmitting the plurality of X-ray beams through apixelated K-edge coded aperture structure, the pixelated K-edge codedaperture structure delineating a plurality of openings, the plurality ofopenings containing at least one pair of balanced K-edge filters;detecting an intensity of the encoded plurality of X-ray beams from theat least one pass; and reconstructing a spectral CT image of the targetfrom the encoded plurality of X-ray beams.
 14. The method of claim 13,wherein the restructuring of the spectral CT image of the target uses aquasi-monochromatic intensity.
 15. The method of claim 14, wherein theat least one pass of the X-ray beam generator comprises a first pass anda second pass.
 16. The method of claim 13, wherein the scanningcomprises either rotating the target, while maintaining an X-raydetector for detecting the intensity of the encoded plurality of X-rayand the X-ray beam generator stationary or rotating the X-ray beamgenerator and rotating an X-ray detector for detecting the intensity ofthe encoded plurality of X-ray beams.
 17. The method of claim 13,wherein the plurality of X-ray beams are generated with a plurality ofX-ray generators, each X-ray generator associated with a respectivepixelated K-edge coded aperture structure.
 18. A system for generating aspectral computed tomography, the system comprising: at least one X-raygenerator configured to produce a plurality of X-ray beams; at least onepixelated K-edge coded aperture structure delineating a plurality ofopenings, the plurality of openings associated with at least one K-edgefilter, the plurality of openings configured to receive and transmit theplurality of X-ray beams with a difference in an energy bandcorresponding to a difference between K-edge values of a correspondingbalanced pair of K-edge filters; and at least one detector configured todetect the plurality of X-ray beams transmitted through the at least onepixelated K-edge coded aperture structure.
 19. The system of claim 18,wherein the at least one pixelated K-edge aperture has a non-randomizedpattern and is configured to encode the plurality of X-ray beamstransmitted through the at least one pixelated K-edge coded aperturestructure with the non-randomized pattern.
 20. The system of claim 18,wherein each of the plurality of openings of the at least one pixelatedK-edge aperture is arranged with respect to other openings to spatiallyencode the plurality of X-ray beams transmitted through the plurality ofopenings of the at least one pixelated K-edge coded aperture structure.21. The system of claim 18, wherein the at least one detector is a linedetector or a two dimensional detector.
 22. The system of claim 18,wherein the detector elements are positioned on a semicircular (arch)geometry.
 23. The system of claim 18, further comprising a plurality ofX-ray generators, each X-ray generator associated with a respectivepixelated K-edge coded aperture structure.
 24. The system of claim 18,wherein a geometry of the system corresponds to a tomosynthesis system.25. The system of claim 18, wherein the at least one X-ray generatorsand the at least one detector are rotatable.
 26. The system of claim 18,further comprising a patient positioning system configured forpositioning a target.